# Complex Fractions

A fraction that contains one or more fractions in its numerator and denominator is called a complex fraction.

Examples are:
$\frac{{\frac{3}{{x{y^2}}}}}{{\frac{2}{{{x^2}y}}}},\,\frac{{\frac{x}{y} – \frac{y}{x}}}{{\frac{1}{x} + \frac{1}{y}}}\,and\,\frac{{{x^2} – \frac{1}{x}}}{{x + \frac{1}{x} + 1}}$

A complex fraction may be simplified by reducing its numerator and denominator (separately) to simple fractions and then dividing.

Example:

Simplify the complex fraction $\frac{{1 + \frac{1}{x}}}{{x – \frac{1}{x}}}$

Solution:

$\frac{{1 + \frac{1}{x}}}{{x – \frac{1}{x}}} = \frac{{\frac{{x + 1}}{x}}}{{\frac{{{x^2} – 1}}{x}}} = \frac{{x + 1}}{x} \div \frac{{{x^2} – 1}}{x} = \frac{{x + 1}}{x} \cdot \frac{x}{{{x^2} – 1}}$
$= \frac{{\left( {x + 1} \right)x}}{{\left( {{x^2} – 1} \right)x}} = \frac{{x + 1}}{{\left( {x – 1} \right)\left( {x + 1} \right)}} = \frac{1}{{x – 1}}$

An alternative method for simplifying a complex fraction is to multiply its numerator and denominator by the LCD of all fractions occurring in its numerator and denominator. The resulting fraction may then be simplified by cancellation.

Example:

Simplify the complex fraction $\frac{{\frac{1}{{{x^3}}} + \frac{2}{{{x^2}y}} + \frac{1}{{x{y^2}}}}}{{\frac{y}{{{x^2}}} – \frac{1}{y}}}$

Solution:

The LCD of ${x^3}$, ${x^2}y$, $x{y^2}$, ${x^2}$ and $y$ is ${x^3}{y^2}$.

Therefore
$\frac{{\frac{1}{{{x^3}}} + \frac{2}{{{x^2}y}} + \frac{1}{{x{y^2}}}}}{{\frac{y}{{{x^2}}} – \frac{1}{y}}} = \frac{{{x^3}{y^2}\left( {\frac{1}{{{x^3}}} + \frac{2}{{{x^2}y}} + \frac{1}{{x{y^2}}}} \right)}}{{{x^3}{y^2}\left( {\frac{y}{{{x^2}}} – \frac{1}{y}} \right)}} = \frac{{{y^2} + 2xy + {x^2}}}{{x{y^2} – {x^3}y}}$
$= \frac{{{{\left( {y + x} \right)}^2}}}{{xy\left( {{y^2} – {x^2}} \right)}} = \frac{{{{\left( {y + x} \right)}^2}}}{{xy\left( {y – x} \right)\left( {y + x} \right)}} = \frac{{y + x}}{{xy\left( {y – x} \right)}}$